Research On The Self-shrinking Solutions Of Mean Curvature Flow

Mean curvature flow evolves hypersurfaces in their normal direction with speed equal to the mean curvature at each point.One of the most important problems in mean curvature flow is to understand the possible singularities that the flow goes through.Singularities are unavoidable as the flow contracts any closed embedded hypersurface in Euclidean space eventually leading to extinction of the evolving hypersurface.A hy-persurface is a self-shrinker if it satisfies the equation H-(X,N>/2=0.Self-shrinkers play an important role in the study of the mean curvature flow.Not only do they corre-spond to self-shrinking solutions to the mean curvature flow,but also they describe all possible blow ups at a given singularity of the mean curvature flow.It is well known that the L-operator plays an extremely important role in the study of self-shrinkers.In this paper,we investigate the positive solutions of three equations Lu = 0,Lu=(?),and Lu = Au on self-shrinkers.And we prove the global gradient estimate and local gradient estimate for the positive solutions.Then we collect some applications of the gradient estimates for the positive solutions on self-shrinkers,such as lower estimates for the first non-zero eigenvalue for the differential operator L,Hamack inequalities and so on.Next,we derive a Reilly inequality for drifting Laplacian operator on self-shrinkers of the mean curvature flow.Using this Reilly inequality,we obtain some new Poincare inequalities not only on a self-shrinker,but more interestingly,also on its boundary.Moreover,we get some eigenvalue estimates for the drifting Laplacian operator on a compact self-shrinker and its boundary,and also make some global esti-mates on the generalized mean curvature Hp on the boundary.As a generalization of generic mean curvature flow,we also consider the volume-preserving mean curvature flow,and investigate the critical points of the weighted area functional for the weight-ed volume-preserving variations,?-hypersurfaces satisfying H-<X,N>/2= ?.First of all,we study the classification and rigidity of A-hypersurfaces without the assump-tion on polynomial volume growth in the weighted volume-preserving mean curvature flow,and generalize many meaningful results of self-shrinkers to A-hypersurfaces by the generalized maximum principle for L-operator,the weighted L2-condition on the norm of the second fundamental form,Sobolev inequality on hypersurface and the p-parabolicity of ?-hypersurfaces.In addition,we investigate the volume comparison theorem of complete bounded ?-hypersurfaces with |A|?? and get some application-s of the volume comparison theorem,such as,Milnor theorem and Hurewicz theorem.Next,we consider the relation among A,extrinsic radius k,intrinsic diameter d.,and dimension n of the complete ?-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius.At last,we get some topological properties of the bounded ?-hypersurface with some natural and general restrictions.